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Chapter 2: Problem 19

Use the precise definition of a limit to prove the following limits. $$\lim _{x \rightarrow 1}(8 x+5)=13$$

Short Answer

Expert verified
Question: Prove using the precise definition of a limit that $$\lim_{x \rightarrow 1} (8x + 5) = 13$$. Answer: Given any \(\epsilon>0\), we found that there exists a \(\delta = \frac{\epsilon}{8}\) such that if \(0 <|x-1|<\delta\), then \(|8x - 8| < \epsilon\). Therefore, the limit of the function \(f(x) = 8x + 5\) as \(x\) approaches 1 is 13.

Step by step solution

01

Understand the precise definition of the limit and write it down

First, let's write down the precise definition of the limit for this problem: Given any \(\epsilon>0\), there exists a \(\delta > 0\) such that if \(0<|x-1|<\delta\), then \(|(8x + 5)-13|<\epsilon\).
02

Simplify the inequality

Now, simplify the inequality under the limit condition: \(|(8x + 5) - 13| < \epsilon\) \(|8x - 8| < \epsilon\).
03

Factor the expression

We can factor out an 8 in the absolute value expression: \(8|x-1| < \epsilon\).
04

Find the value of \(\delta\) in terms of \(\epsilon\)

We need to find a value for \(\delta\) that ensures the inequality is true. In this case, we can set \(\delta = \frac{\epsilon}{8}\). Plug \(\delta\) back into the inequality: \(0 < |x-1| < \frac{\epsilon}{8}\). Since this inequality holds, we can conclude that: $$\lim_{x \rightarrow 1} (8x+5) = 13$$. Hence, we have proven the limit using the precise definition of the limit.

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